# Regular category

In category theory, a **regular category** is a category with finite limits and coequalizers of a pair of morphisms called **kernel pairs**, satisfying certain *exactness* conditions. In that way, regular categories recapture many properties of abelian categories, like the existence of *images*, without requiring additivity. At the same time, regular categories provide a foundation for the study of a fragment of first-order logic, known as regular logic.

A category *C* is called **regular** if it satisfies the following three properties:^{[1]}

In a regular category, the regular-epimorphisms and the monomorphisms form a factorization system. Every morphism *f:X→Y* can be factorized into a regular epimorphism *e:X→E* followed by a monomorphism *m:E→Y*, so that *f=me*. The factorization is unique in the sense that if *e':X→E' *is another regular epimorphism and *m':E'→Y* is another monomorphism such that *f=m'e'*, then there exists an isomorphism *h:E→E' * such that *he=e' *and *m'h=m*. The monomorphism *m* is called the **image** of *f*.

A functor between regular categories is called **regular**, if it preserves finite limits and coequalizers of kernel pairs. A functor is regular if and only if it preserves finite limits and exact sequences. For this reason, regular functors are sometimes called **exact functors**. Functors that preserve finite limits are often said to be **left exact**.

Regular logic is the fragment of first-order logic that can express statements of the form

which is natural in *C*. Here, *R(T)* is called the *classifying* category of the regular theory *T.* Up to equivalence any small regular category arises in this way as the classifying category of some regular theory.^{[2]}

A regular category is said to be **exact**, or **exact in the sense of Barr**, or **effective regular**, if every equivalence relation is effective.^{[4]} (Note that the term "exact category" is also used differently, for the exact categories in the sense of Quillen.)